# Who discovered the rational root theorem?

As the title says, I would like to know who discovered the rational root theorem. The Encyclopaedia Britannica states that “The 17th-century French philosopher and mathematician René Descartes is usually credited with devising the test”, but I was unable to find any reference to this both in A History of Algebra: From al-Khwārizmī to Emmy Noether (by van der Waerden) or in Taming the Unknown: A History of Algebra from Antiquity to the Early Twentieth Century (by Victor J. Katz and Karen Hunger Parshall).

## 1 Answer

When in doubt, go to the source. There is a nice facsimile edition of Descartes's La Géométrie with English translation, freely available on Internet Archive. Descartes does not spell out the trick as a theorem, to which it was elevated by modern textbooks, or even as a general rule, but he does use it in several places, and in a generalized form to cover quadratic irrationals. For example on p.176 he writes

"Given, for example, $$y^6—8y^4—124y^2—64=0$$. The last term, 64, is divisible by 1, 2, 4, 8, 16, 32, and 64; therefore we must find whether the left member is divisible by $$y^2—1$$, $$y^2+1$$, $$y^2—2$$, $$y^2+2$$, $$y^2—4$$, and so on."

And on p. 180 explains

"After removing any surds or fractions, see if a binomial having one term a factor of the last term of the expression will divide the left member. If such a binomial can be found, either the known quantity of the binomial is the required root, or, after the division is performed, the resulting equation, which is of only three dimensions, must be treated in the same way."

He does not bother with rational roots and divisors of the leading coefficient because on pp. 172-175 he explains a trick "useful in changing fractional terms of an equation to whole numbers, and often in rationalizing the terms", which is illustrated by transforming an equation to convert its fractional roots into integers (and quadratically irrational roots into rational first).

Historians did not make much of it either. For example, Boyer in Descartes and the Geometrization of Algebra only mentions it in passing:

"That so much of La Géométrie is concerned with the theory of equations — the number of possible positive roots, increasing and decreasing the roots of an equation, finding rational roots, and depressing the degree when a root is known — does not indicate a preference for algebra over geometry."

Serfati is equally brief:

"He also proves that for a polynomial $$P$$ to be divisible by $$(X − a)$$ it is necessary and sufficient that $$P(a)$$ be zero. He then uses his indeterminate coefficients to describe the division of a polynomial by $$(X−a)$$. So it was important for him to know at least one root. For an equation with rational coefficients, he studies the rational roots, if any."